Composite Quantum Systems and Entanglement
Stub. The multiple-system axiom: combining quantum systems via the tensor product rather than a Cartesian product, which is the single mathematical fact responsible for everything distinctively quantum about multi-party systems — including entanglement, introduced here via product states and the states that aren't product states.
Corresponds to Lectures 4–5. Classically, combining two systems multiplies their state-space sizes; quantum mechanically, combining two systems tensors their state spaces, and a tensor product space is far bigger than any classical product — most of its vectors aren’t expressible as a simple pairing of one state per subsystem. States that are such a pairing are product states; the rest, including the canonical Bell states, are entangled — correlated in a way with no classical analogue, since measuring one subsystem instantaneously affects what can be said about the other, however far apart they are. This first exposure to entanglement is purely at the level of pure states and product-vs-not; quantifying and detecting entanglement in the general (mixed-state) case is the subject of the-separability-problem-and-entanglement-witnesses much later in the course.
Outline (TODO — flesh out each)
- The multiple system axiom — tensor product spaces
- Why tensor, not Cartesian, product
- Product states
- Entanglement — a first look, Bell states